Curve25519

In cryptography, Curve25519 is an elliptic curve offering 128 bits of security and designed for use with the elliptic curve Diffie–Hellman (ECDH) key agreement scheme. It is one of the fastest ECC curves; it is not covered by any known patents,^[1] and it is less susceptible to weak random-number generators. The reference implementation is public domain software.^[2]

The original Curve25519 paper defined it as a Diffie–Hellman (DH) function. Daniel J. Bernstein has since proposed that the name Curve25519 be used for the underlying curve, and the name X25519 for the DH function.^[3]

Mathematical properties

The curve used is y² = x³ + 486662x² + x, a Montgomery curve, over the prime field defined by the prime number 2²⁵⁵ − 19, and it uses the base point x = 9. The protocol uses compressed elliptic point (only X coordinates), so it allows for efficient use of the Montgomery ladder for ECDH, using only XZ coordinates.^[4]

Curve25519 is constructed such that it avoids many potential implementation pitfalls.^[5] By design, it avoids many side channel attacks and issues with poor-quality random-number-generators.

The curve is birationally equivalent to Ed25519, a Twisted Edwards curve.^[6]

Popularity

Curve25519 was first released by Daniel J. Bernstein in 2005,^[7] but interest increased considerably after 2013 when it was discovered that the NSA had implemented a backdoor into Dual_EC_DRBG. While not directly related,^[8] suspicious aspects of the NIST's P curve constants^[9] led to concerns^[10] that the NSA had chosen values that gave them an advantage in factoring^[11] public keys.^[12]

I no longer trust the constants. I believe the NSA has manipulated them through their relationships with industry

— Bruce Schneier, The NSA Is Breaking Most Encryption on the Internet (2013)

Since then, Curve25519 has become the de facto alternative to P-256, and is used in a wide variety of applications.^[13] In 2014 OpenSSH^[14] defaults to Curve25519-based ECDH.

Libraries

• Libgcrypt^[15]
• libssh^[14][16]
• NaCl^[17]
• GnuTLS^[18]
• mbed TLS (formerly PolarSSL)^[19]
• wolfSSL^[20]
• Botan^[21]
• Libsodium^[22]
• OpenSSL since version 1.1.0^[23]
• NaCl for Tcl — a port to the Tcl language.^[24]

Applications

• DNSCrypt^[25]
• DNSCurve
• Dropbear^[16][26]
• GNUnet^[27]
• GnuPG
• I2P^[28]
• iOS^[29]
• OMEMO, a proposed extension for XMPP (Jabber)^[30]
  • Conversations (Android)
  • Cryptocat (Linux, OS X, Windows)^[31]
  • Gajim via plugin (BSD, Linux, Windows)^[32]
• minilock^[33]
• OpenBSD (used to sign releases and packages^[34][35])
• OpenSSH^[16]
  • Exclusive key exchange in OpenSSH 6.7 when compiled without OpenSSL.^[36][37]
• Peerio^[38]
• RLogin^[16]
• Signal Protocol
  • Facebook Messenger
  • Google Allo
  • Signal
  • WhatsApp
• Silent Phone
• SmartFTP^[16]
• SSHJ^[16]
• Threema Instant Messenger^[39]
• TinySSH^[16]
• TinyTERM^[16]
• Tor^[40]
• Tox
• Viber^[41]
• Wire

See also

• Ed25519

References

External links

• Official website